MATLAB code and output files for integral, mean and covariance of the simplex-truncated multivariate normal distribution

Compositional data, which is data consisting of fractions or probabilities, is common in many fields including ecology, economics, physical science and political science. If these data would otherwise be normally distributed, their spread can be conveniently represented by a multivariate normal distribution truncated to the non-negative space under a unit simplex. Here this distribution is called the simplex-truncated multivariate normal distribution. For calculations on truncated distributions, it is often useful to obtain rapid estimates of their integral, mean and covariance; these quantities characterising the truncated distribution will generally possess different values to the corresponding non-truncated distribution.

In the paper Adams, Matthew (2022) Integral, mean and covariance of the simplex-truncated multivariate normal distribution. PLoS One, 17(7), Article number: e0272014. https://eprints.qut.edu.au/233964/, three different approaches that can estimate the integral, mean and covariance of any simplex-truncated multivariate normal distribution are described and compared. These three approaches are (1) naive rejection sampling, (2) a method described by Gessner et al. that unifies subset simulation and the Holmes-Diaconis-Ross algorithm with an analytical version of elliptical slice sampling, and (3) a semi-analytical method that expresses the integral, mean and covariance in terms of integrals of hyperrectangularly-truncated multivariate normal distributions, the latter of which are readily computed in modern mathematical and statistical packages. Strong agreement is demonstrated between all three approaches, but the most computationally efficient approach depends strongly both on implementation details and the dimension of the simplex-truncated multivariate normal distribution.

This dataset consists of all code and results for the associated article.

组合数据，即由分数或概率组成的数据，在生态学、经济学、物理科学和政治科学等多个领域颇为常见。若此类数据原本呈正态分布，其分布范围可便捷地通过限制于单位单纯形下的非负空间的多元正态分布来表示。在此，此类分布被称为单纯形截断多元正态分布。对于截断分布的计算，获得其积分、均值和协方差的快速估计通常颇具价值；这些表征截断分布的量通常与相应的非截断分布具有不同的数值。在Adams, Matthew (2022)发表的论文《单纯形截断多元正态分布的积分、均值和协方差》中（PLoS One, 17(7), 文章编号：e0272014. https://eprints.qut.edu.au/233964/），描述并比较了三种可以估计任何单纯形截断多元正态分布的积分、均值和协方差的不同方法。这三种方法分别是：（1）简单拒绝抽样，（2）由Gessner等人描述的一种方法，该方法将子集模拟、Holmes-Diaconis-Ross算法与分析形式的椭圆形切片采样统一起来，（3）一种半解析方法，该方法将积分、均值和协方差表达为超矩形截断多元正态分布的积分，而后者在现代数学和统计软件中易于计算。三种方法之间均表现出高度的一致性，但最有效的计算方法强烈依赖于实现细节以及单纯形截断多元正态分布的维度。本数据集包含了相关文章的所有代码和结果。